#126536

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₄³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₄³
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1726
prime radical	C₂³	#405
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,3,6,4,5,7,8)(9,10,13,20,15,23,40,21,17,22,41,55,16,24,14,42)(11,25,27,45,29,18,28,44,31,47,38,66,30,46,63,53)(12,32,33,65,35,51,34,54,37,52,39,50,36,19,48,49)(26,43,57,67,59,64,71,68,61,69,62,70,60,56,58,72);(9,11,14,28,16,30,41,27,17,31,40,63,15,29,13,38)(10,18,21,45,24,25,20,53,22,46,42,66,23,47,55,44)(12,26,34,58,36,60,33,62,37,61,48,71,35,59,39,57)(19,43,50,68,52,56,54,67,51,69,65,72,32,64,49,70);(2,5)(6,8)(9,12)(10,19,22,51)(11,26)(13,39)(14,34)(15,35)(16,36)(17,37)(18,43,46,69)(20,54,55,49)(21,50,42,65)(23,32,24,52)(25,56,47,64)(27,62)(28,58)(29,59)(30,60)(31,61)(33,41)(38,57)(40,48)(44,70,53,67)(45,68,66,72)(63,71)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bh,c^a=cg,e^a=ei,f^a=fi,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33732);